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Local Maxima of the Potential Energy on Spheres

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Let S d be a unit sphere in ℝd+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N S d, we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ ij ||x i x j ||α. The following theorem is proved: for αd − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima.

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References

  1. 1.

    A. Bondarenko and M. Viazovska, “Spherical designs via Brouwer fixed point theorem,” SIAM J. Discrete Math., 24, 207–217 (2010).

  2. 2.

    H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres,” J. Amer. Math. Soc., 20, No. 1, 99–148 (2007).

  3. 3.

    E. B. Saff and A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intelligencer, 19, No. 1, 5–11 (1997).

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Author information

Correspondence to D. V. Radchenko.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 10, pp. 1427–1429, October, 2013.

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Radchenko, D.V. Local Maxima of the Potential Energy on Spheres. Ukr Math J 65, 1585–1587 (2014). https://doi.org/10.1007/s11253-014-0880-4

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Keywords

  • Potential Energy
  • Local Maximum
  • Point Theorem
  • Unit Sphere
  • Positive Real Number